#7625055

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#62495	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18,21,26)(19,28,31,44)(20,25,23,34)(22,36,35,39)(24,38,41,32)(27,29,42,37)(30,43,33,48)(40,47,45,46);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,18)(19,24)(20,34)(21,26)(22,27)(23,25)(28,32)(29,39)(30,45)(31,41)(33,40)(35,42)(36,37)(38,44)(43,47)(46,48);(17,19,29,46,23,33,22,32,21,31,37,47,20,30,35,38)(18,24,39,48,25,40,27,28,26,41,36,43,34,45,42,44);(3,10)(6,9)(11,16)(14,15)(17,20,21,23)(18,25,26,34)(19,30,31,33)(22,29,35,37)(24,40,41,45)(27,36,42,39)(28,43,44,48)(32,46,38,47);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,26)(24,41)(25,34)(27,42)(28,44)(36,39)(40,45)(43,48)

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532251);